Orbital Semilattices

TL;DR

This paper introduces orbital semilattices, a mathematical structure combining semilattices with additional operations like outer multiplication and diagonals, and demonstrates their representation via table algebras.

Contribution

It defines orbital semilattices and proves they can be represented as subalgebras of table algebras, linking algebraic structures with relational models.

Findings

Orbital semilattices are a new algebraic structure with specific operations.

Each orbital semilattice can be embedded into a table algebra.

The framework connects algebraic and relational representations.

Abstract

Orbital semilattices are introduced as bounded semilattices that are, in addition, equipped with an outer multiplication (a semigroup action) and diagonals (a concept borrowed from cylindric algebra), where each semilattice element has a certain domain. An example of an orbital semilattice is a table algebra, where the domain is the schema, the diagonals are diagonal relations, and outer multiplication encodes renaming, projection and column duplication. It is shown that each orbital semilattice can be represented by a subalgebra of such a table algebra.